From: hoey@AIC.NRL.Navy.Mil (Dan Hoey)
Newsgroups: sci.math,rec.puzzles
Subject: Results: Unfolding the tesseract
Date: 16 Oct 92 22:57:12 GMT
Sender: usenet@ra.nrl.navy.mil
Followup-To: sci.math
Organization: Naval Research Laboratory, Washington, DC

Back in June, orourke@sophia.smith.edu (Joseph O'Rourke) wrote:

>Does anyone know how many distinct unfoldings there are of a
>hypercube (a 4-dim cube)?  I know there are 11 distinct 2-dim
>connected shapes that can result from unfolding a (3-dim) cube
>[Rucker, "The Fourth Dimension" (1984) p.34].

Well, Rudy Rucker did pass near the topic, but it was covered in
somewhat more detail in Martin Gardner's _Mathematical_Carnival_
[1975] article on the hypercube.  Gardner mentioned that he posed the
question of many ways there are of unfolding a tesseract to
_Scientific_American_ readers, and he got so many answers he couldn't
decide which (if any) was right.

I spent some time this summer counting them.  I organized them and
counted them by hand, and got 253 cases.  Then I reorganized some of
them, and noticed some cases I had missed, and now there were 264.
Then I wrote a program to count them, and came up with 261; I soon
noticed three duplicates in my hand work.  Then I compared the
program's output with my table, case by case, and they matched.  So at
this point I am fairly certain there are exactly 261 ways of unfolding
the surface of a tesseract into an octocube.  (And I am fairly
sympathetic with Garder's readers.)

Gardner noted that the eleven hexominoes you get by unfolding the
surface of a cube:
                                                                  x
     xxx    x     xx    x     xx    xx    x     x     xx     x    x
      x    xxx    x     xx   xx     x     x     xx    x     xx   xx
      x     x     x    xx     x    xx    xxx   xx    xx    xx    x
      x     x    xx     x     x     x    x     x     x     x     x

cannot be used to tile a rectangle.  I do not know if he tried tiling
with the twenty hexominoes you get if you add the reflections of the
mirror-asymmetric hexominoes and forbid turning them over.

As for the 261 octocubes you get from unfolding a tesseract, were we
to build them, it would be infeasible to ``turn them over'' into their
mirror images.  Therefore we would probably prefer to build a
rectangular prism from the 355 octocubes we get by adding mirror
images of the 194 mirror-asymmetric octocubes.  But I have no plans to
check whether that is possible.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil

----------------------------------------------------------------------
From: Torsten Sillke

The number 261 matches the value found by Peter Turney.

References:
  - A. Sanders, D. V. Smith;
    Nets of the octahedron and the cube.
    Mathematics Teaching 42 (Spring 1968) 60-63
    (Finds all 11 nets for the octahedron an shows a duality with the cube.)

  - David Singmaster;
    Sources in Recreational Mathematics,
    An Annotated Bibliography, 6th Pre. Ed., Nov. 1993
    Part I, Sect. 6.AA Nets of Polyhedra, p 195
    (list references: [A. Sanders, D. V. Smith], [P. Turney])

  - Peter Turney;
    Unfolding the tesseract,
    JRM 17 (1984-85) 1-16